| L(s) = 1 | + 15·5-s + 27·11-s − 3·13-s − 15·17-s + 78·19-s + 105·23-s + 150·25-s − 117·29-s + 207·31-s − 120·37-s − 300·41-s + 483·43-s + 303·47-s − 522·49-s + 492·53-s + 405·55-s + 240·59-s − 444·61-s − 45·65-s + 522·67-s − 168·71-s − 876·73-s + 2.10e3·79-s − 42·83-s − 225·85-s + 2.26e3·89-s + 1.17e3·95-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.740·11-s − 0.0640·13-s − 0.214·17-s + 0.941·19-s + 0.951·23-s + 6/5·25-s − 0.749·29-s + 1.19·31-s − 0.533·37-s − 1.14·41-s + 1.71·43-s + 0.940·47-s − 1.52·49-s + 1.27·53-s + 0.992·55-s + 0.529·59-s − 0.931·61-s − 0.0858·65-s + 0.951·67-s − 0.280·71-s − 1.40·73-s + 2.99·79-s − 0.0555·83-s − 0.287·85-s + 2.70·89-s + 1.26·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(11.63558856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.63558856\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 522 T^{2} + 2666 T^{3} + 522 p^{3} T^{4} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 27 T + 2721 T^{2} - 79918 T^{3} + 2721 p^{3} T^{4} - 27 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 3 T + 4962 T^{2} + 32735 T^{3} + 4962 p^{3} T^{4} + 3 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 15 T + 5031 T^{2} - 273298 T^{3} + 5031 p^{3} T^{4} + 15 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 78 T + 9942 T^{2} - 303500 T^{3} + 9942 p^{3} T^{4} - 78 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 105 T + 25413 T^{2} - 2052802 T^{3} + 25413 p^{3} T^{4} - 105 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 117 T + 67683 T^{2} + 5431966 T^{3} + 67683 p^{3} T^{4} + 117 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 207 T + 69741 T^{2} - 8616098 T^{3} + 69741 p^{3} T^{4} - 207 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 120 T - 39264 T^{2} - 10421922 T^{3} - 39264 p^{3} T^{4} + 120 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 300 T + 82023 T^{2} + 31686600 T^{3} + 82023 p^{3} T^{4} + 300 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 483 T + 251961 T^{2} - 76432898 T^{3} + 251961 p^{3} T^{4} - 483 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 303 T + 166137 T^{2} - 65064754 T^{3} + 166137 p^{3} T^{4} - 303 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 492 T + 476259 T^{2} - 138372072 T^{3} + 476259 p^{3} T^{4} - 492 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 240 T + 213705 T^{2} + 18543136 T^{3} + 213705 p^{3} T^{4} - 240 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 444 T + 185052 T^{2} - 860902 T^{3} + 185052 p^{3} T^{4} + 444 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 522 T + 370530 T^{2} - 307525264 T^{3} + 370530 p^{3} T^{4} - 522 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 168 T - 41151 T^{2} - 401172816 T^{3} - 41151 p^{3} T^{4} + 168 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 12 p T + 1148628 T^{2} + 668051142 T^{3} + 1148628 p^{3} T^{4} + 12 p^{7} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 2103 T + 2885448 T^{2} - 2370776439 T^{3} + 2885448 p^{3} T^{4} - 2103 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 42 T + 653901 T^{2} + 416514772 T^{3} + 653901 p^{3} T^{4} + 42 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2268 T + 3125643 T^{2} - 3038479992 T^{3} + 3125643 p^{3} T^{4} - 2268 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1392 T + 3023340 T^{2} + 2460493802 T^{3} + 3023340 p^{3} T^{4} + 1392 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.907188063133290343527553725857, −7.25593137476536045421016334869, −7.16849809364673228016656957760, −6.84567770276326696293066134168, −6.67121043329780711788790803197, −6.38847033710782497743079818975, −6.06687952113027637796009514626, −5.88130746269755240157892364941, −5.45550129367294652911363559206, −5.45218243618999302573208374452, −4.91461584590121560834030244784, −4.77432596857426202244982986955, −4.63368401722123216757467206939, −3.95378240751316333405268748631, −3.81460803817082611539188554412, −3.59049683057299957096515029460, −2.92084131241598527841500768769, −2.91938802305666395474060611604, −2.62028168400331556811979886254, −1.92487551084706661762631249280, −1.90568396701408797882835132096, −1.57633405942333002231355928279, −0.958356064454058177720609227916, −0.74320694347671143829968336529, −0.48418247747877377240088489730,
0.48418247747877377240088489730, 0.74320694347671143829968336529, 0.958356064454058177720609227916, 1.57633405942333002231355928279, 1.90568396701408797882835132096, 1.92487551084706661762631249280, 2.62028168400331556811979886254, 2.91938802305666395474060611604, 2.92084131241598527841500768769, 3.59049683057299957096515029460, 3.81460803817082611539188554412, 3.95378240751316333405268748631, 4.63368401722123216757467206939, 4.77432596857426202244982986955, 4.91461584590121560834030244784, 5.45218243618999302573208374452, 5.45550129367294652911363559206, 5.88130746269755240157892364941, 6.06687952113027637796009514626, 6.38847033710782497743079818975, 6.67121043329780711788790803197, 6.84567770276326696293066134168, 7.16849809364673228016656957760, 7.25593137476536045421016334869, 7.907188063133290343527553725857
